372 



MEMOIRS OF THE AMERICAN ACADEMY. 



nothing, 

 that it i 



If 



be a logical species, then, it is necessary to suppose 



addition 



and further that some v is y 



In short 



necessary to assume 



the truth of a proposition, which, being itself particular, presents the 



difficulty in regard to its symbolical expression 



Moreover, from 



v,y 



v,(l 



we can, according to algebraic principles, deduce successively 



v,y 



V 



\,x 



v,x 



V 



v,y 



v,(l 



Now if the first equation means that some Y's are not X's, the last ought to mean 

 that some X's are not Y's ; for the algebraic forms are the same, and the question is, 

 whether the algebraic forms are adequate to the expression of particulars. It would 

 appear, therefore, that the inference from Some Y's are not X's to Some X's are not 

 Y's, is good ; but it is not so, in fact. 



What is wanted, in order to express hypothetical and particulars analytically, is a 

 relative term which shall denote " case of the existence of — ," or " what exists only 

 if there is any — • n ; or else " case of the non-existence of — ," or " what exists only 



if there 



»> 



When Boole's algebra is extended to relative terms, it is easy 



that having expressed 



see what these particular relatives must be. For suppose 



the propositions " it thunders," and " it lightens," we wish to express the fact that 



" if it lightens, it thunders." Let 



A 







and 



B 



0, 



be equations meaning respectively, it lightens and it thunders, 

 when x does not and vice versa, whatever x may be, the formula 



Then, if (px vanishes 



A 



cpB 



* 



expresses that if it lightens it thunders; for if it lightens, A vanishes; hence <pA does 

 not vanish, hence <pB does not vanish, hence B vanishes, hence it thunders. It rnake^- 

 no difference what the function <p is, provided only it fulfils the condition mentioned. 

 Now, 0* is such a function,, vanishing when x does not, and not vanishing when x does. 

 Zero, therefore, may be interpreted as denoting " that which exists if, and only if, 



th ere 



» 



Then the equation 



0° 



1 



